Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma
The aim of this work is the evaluation of thermodynamic and transport properties of fusion plasmas in an externally applied strong magnetic field (10…103 T). For these purpose the Thomas-Fermi model for substances with a given temperature and density was used. The effect of such strong magnetic fiel...
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Kuzenov, V.V. Ryzhkov, S.V. Shumaev, V.V. 2015-05-25T05:55:58Z 2015-05-25T05:55:58Z 2015 Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma / V.V. Kuzenov, S.V. Ryzhkov, V.V. Shumaev // Вопросы атомной науки и техники. — 2015. — № 1. — С. 97-99. — Бібліогр.: 19 назв. — англ. 1562-6016 PACS: 52.25.Fi, 52.25.Kn, 52.25.Xz, 52.65.-y https://nasplib.isofts.kiev.ua/handle/123456789/82098 The aim of this work is the evaluation of thermodynamic and transport properties of fusion plasmas in an externally applied strong magnetic field (10…103 T). For these purpose the Thomas-Fermi model for substances with a given temperature and density was used. The effect of such strong magnetic field on the transport properties of plasmas and the view of the inner shells of atoms and ions was analyzed. Isotherms of the pressure and the specific internal energy of the tungsten plasma are obtained. Quantum and exchange corrections to the pressure of this plasma have been taken into account. Определены термодинамические и транспортные свойства термоядерной плазмы, находящейся в сильном внешнем магнитном поле (10…1000 Тл). Для этой цели использовалась модель Томаса-Ферми для веществ с заданным температурой и плотностью. Проанализировано действие сильного магнитного поля на транспортные свойства плазмы и вид внутренних оболочек атомов и ионов. Построены изотермы давления, удельной внутренней энергии и энтропии плазмы вольфрама. Учтены квантовые и обменные поправки к давлению плазмы. Визначено термодинамічні і транспортні властивості термоядерної плазми, що знаходиться в сильному зовнішньому магнітному полі (10...1000 Тл). Для цієї мети використовувалася модель Томаса-Фермі для речовин із заданною температурою і щільністю. Проаналізовано дію сильного магнітного поля на транспортні властивості плазми і вид внутрішніх оболонок атомів і іонів. Побудовані ізотерми тиску, питомої внутрішньої енергії і ентропії плазми вольфраму. Враховані квантові і обмінні поправки до тиску плазми. This work was supported by the Ministry of Education and Science of the Russian Federation (Goszadanie № 13.79.2014/K). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma Применение модели Томаса-Ферми для определения термодинамических свойств замагниченной плазмы Застосування моделі Томаса-Фермі для визначення термодинамічних властивостей замагніченої плазми Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma |
| spellingShingle |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma Kuzenov, V.V. Ryzhkov, S.V. Shumaev, V.V. Фундаментальная физика плазмы |
| title_short |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma |
| title_full |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma |
| title_fullStr |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma |
| title_full_unstemmed |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma |
| title_sort |
application of thomas-fermi model to evaluation of thermodynamic properties of magnetized plasma |
| author |
Kuzenov, V.V. Ryzhkov, S.V. Shumaev, V.V. |
| author_facet |
Kuzenov, V.V. Ryzhkov, S.V. Shumaev, V.V. |
| topic |
Фундаментальная физика плазмы |
| topic_facet |
Фундаментальная физика плазмы |
| publishDate |
2015 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Применение модели Томаса-Ферми для определения термодинамических свойств замагниченной плазмы Застосування моделі Томаса-Фермі для визначення термодинамічних властивостей замагніченої плазми |
| description |
The aim of this work is the evaluation of thermodynamic and transport properties of fusion plasmas in an externally applied strong magnetic field (10…103 T). For these purpose the Thomas-Fermi model for substances with a given temperature and density was used. The effect of such strong magnetic field on the transport properties of plasmas and the view of the inner shells of atoms and ions was analyzed. Isotherms of the pressure and the specific internal energy of the tungsten plasma are obtained. Quantum and exchange corrections to the pressure of this plasma have been taken into account.
Определены термодинамические и транспортные свойства термоядерной плазмы, находящейся в сильном внешнем магнитном поле (10…1000 Тл). Для этой цели использовалась модель Томаса-Ферми для веществ с заданным температурой и плотностью. Проанализировано действие сильного магнитного поля на транспортные свойства плазмы и вид внутренних оболочек атомов и ионов. Построены изотермы давления, удельной внутренней энергии и энтропии плазмы вольфрама. Учтены квантовые и обменные поправки к давлению плазмы.
Визначено термодинамічні і транспортні властивості термоядерної плазми, що знаходиться в сильному зовнішньому магнітному полі (10...1000 Тл). Для цієї мети використовувалася модель Томаса-Фермі для речовин із заданною температурою і щільністю. Проаналізовано дію сильного магнітного поля на транспортні властивості плазми і вид внутрішніх оболонок атомів і іонів. Побудовані ізотерми тиску, питомої внутрішньої енергії і ентропії плазми вольфраму. Враховані квантові і обмінні поправки до тиску плазми.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/82098 |
| citation_txt |
Application of Thomas-Fermi model to evaluation of thermodynamic properties of magnetized plasma / V.V. Kuzenov, S.V. Ryzhkov, V.V. Shumaev // Вопросы атомной науки и техники. — 2015. — № 1. — С. 97-99. — Бібліогр.: 19 назв. — англ. |
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2025-11-27T00:24:50Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 97-99. 97
APPLICATION OF THOMAS-FERMI MODEL TO EVALUATION
OF THERMODYNAMIC PROPERTIES OF MAGNETIZED PLASMA
V.V. Kuzenov
1,2
, S.V. Ryzhkov
1
, V.V. Shumaev
1
1
Bauman Moscow State Technical University, Moscow, Russia;
2
A.Yu. Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
E-mail: chubchic@gmail.com
The aim of this work is the evaluation of thermodynamic and transport properties of fusion plasmas in an
externally applied strong magnetic field (10…10
3
T). For these purpose the Thomas-Fermi model for substances
with a given temperature and density was used. The effect of such strong magnetic field on the transport properties
of plasmas and the view of the inner shells of atoms and ions was analyzed. Isotherms of the pressure and the
specific internal energy of the tungsten plasma are obtained. Quantum and exchange corrections to the pressure of
this plasma have been taken into account.
PACS: 52.25.Fi, 52.25.Kn, 52.25.Xz, 52.65.-y
INTRODUCTION
Thermodynamic and transport properties of fusion
plasmas, consisting of the atomic mixture of substances,
in an externally applied strong magnetic field
(10…10
3
T) are important. Such plasma may be used as
cylindrical [1] and spherical [2, 3] target for magnetized
target fusion or magneto-inertial fusion [4-6]. Note that
the degeneracy of the electron gas in such plasmas is
possible [7].
Transport properties of the plasma are the
coefficients of thermal and electrical conductivity.
Thermodynamic properties are described by thermal and
caloric equations of state: P = P(T, ρ), E = E(T, ρ),
S = S(T, ρ), where P is the pressure of the plasma; E, S
are its internal energy and entropy per unit mass. T and
ρ are the temperature and the density of the plasma.
The approximate method is used for the
mathematical description of plasmas thermophysical
properties. This method is based on the generalization
of statistical Thomas-Fermi method to the case of zero
temperature [7] and the external magnetic field [8]. This
method is simple (for example, compared with the
Hartree-Fock-Slater method), while it provides
acceptable accuracy for practice, especially when the
quantum oscillatory and exchange corrections are taken
into account [9].
Currently, various software packages and databases
for determining the thermophysical properties of
substances have been developed: ASTEROID (IPMech
RAS), SESAME (LANL, LLNL Livermore, and
Sandia), TEFIS (KIAM RAS), IVTANTERMO (JIHT
RAS), TERMOS (KIAM), program complexes «TUR»
(RFNC-VNIITF) and EIP EOS (Ioffe Institute) and etc.
The Thomas-Fermi (T-F) model with the quantum and
exchange corrections is used in many of these databases
as a theoretical model. However, some of these
databases are for private use only or they are
incomplete. Also in these databases (except the program
complex EIP EOS [10]) T-F model is not extended to
the case of the external magnetic field impact on
thermonuclear plasma.
1. THE THOMAS-FERMI MODEL:
DESCRIPTION AND APPLICATIONS
The physical model of matter, which is the basis of
the T-F model, assumes that the difference between
"free" and "bound" electrons is absent and a substance is
considered to be composed not of ions and electrons,
but of nuclei and electrons. The interaction energy of
the particles in matter is determined by the electrons.
Calculations of fusion plasmas thermodynamic
properties are based on the model of local
thermodynamic equilibrium (LTE). Systems for a large
number of noninteracting nuclei comply with
Boltzmann statistics.
The calculation of electronic parts of the energy and
the pressure is based on the LTE model, according to
which a substance is separated into the system of atomic
cells; each contains Z electrons (e is the electron charge)
and a nucleus of charge Z∙e. For simplicity, the shape of
the atomic cell is taken to be spherical.
Electrons in the atomic cell are considered as a gas
in the slowly varying radially self-consistent
electrostatic field V(r), generated by the nuclear charge
and the charge of electrons. Thus, the nonideal electron
gas is taken into account. The Fermi-Dirac statistics is
applied to the electron-ion gas.
Assume, that the T-F potential distribution φ(x) =
x∙(V(r)+μ)/θ is known. Here θ is the kB∙T (kB is the
Boltzmann constant); μ is the chemical potential of the
plasma. Then the pressure of electrons Pe at the atomic
cell boundary can be calculated as the average
momentum carried by them per unit e per unit area
through the atomic cell with the radius r0.
It is necessary to take into account the pressure
created by the nuclei to find the total pressure of the
system of particles in the atomic cell. At high
temperatures the system (gas, consisting of cores) is
typically treated as an ideal gas. Therefore the total
pressure P is defined as
θ
ν
eP
, where v is the
volume of the atomic cell.
98 ISSN 1562-6016. ВАНТ. 2015. №1(95)
For details (e.g. for calculating the specific internal
energy and the entropy of the plasma), refer to the [9,
11].
Quantum and exchange corrections are required to
consider because the T-F model is the approximate
method (for details, refer to the [12]). To find them we
use the method, described here [9].
2. ESTIMATIONS OF THE EXTERNAL
MAGNETIC FIELD INFLUENCE ON
THERMOPHYSICAL AND TRANSPORT
PROPERTIES OF PLASMAS
The degree of the magnetic field influence on plasma
transport coefficients (the electrical conductivity, the
thermal conductivity) depends on the ratio of the
collision frequency of electrons ev to the cyclotron
frequency 111,759 10
e
e
e
B B
m
of electrons [13],
here, B is the magnetic induction in T, me is the mass of
the electron.
The magnetic field will have a noticeable effect on
the transport properties of plasmas if 1e ev . We
obtain the condition of the strong magnetic field
influence on the properties of plasmas:
16
3/2
2 10
10
e
Zn
B
T
,
where Z is the ion charge; Λ is the Coulomb logarithm;
Те is the electron temperature in K; n is the plasma
density in m
-3
.
For the tungsten plasma (Z = 74) with parameters of
our interest Те ~ 10
7
K, n ~ 10
25
m
-3
which corresponds
to the plasma density ρ ~ 10 kg/m
3
– crown density of
the target in the inertial confinement fusion) we have
the following estimate (Λ ≈ 15) 7 TB .
In the magneto-inertial fusion it is experimentally
obtained magnetic field Bfus ~ 10
3
T and it can be
achieved even higher values [14-19]. Therefore it is
required to consider the influence of strong magnetic
fields on the transport properties of plasmas.
The magnetic field affects the orientation of spins of
electrons or atoms in the gas which has a temperature T
defined by the condition
or 1,49
B
B k T B T ,
where μ is the Bohr magneton.
In our case, the characteristic temperature is
Тchar ~ 10
7
K. Then 710 TB , that is significantly more
than values reached in the thermonuclear fusion.
The magnetic field B ~ 10
5
T, in which the energy of
the magnetic moment μB is larger than the characteristic
energy of the atom or molecule (It has the order of the
number Ry = mee
4
/2ћ
2
) significantly affects the structure
of atoms and molecules and strongly modifies their
binding energy and ionization energy. Thus, we assume
that the magnetic field does not affect the orientation of
spins of electrons or atoms in a gas. Considered fields
are also much smaller than the field В ~ 10
9
T, so we
can neglect the relativistic effects.
3. RESULTS
The boundary value problem, which determines the
V(r) potential distribution, is solved by the sweeping
method with iterations [9, 11].
The pressure P(T, ρ), the specific internal energy
E(T, ρ) and the entropy S(T, ρ) of the tungsten plasma
are shown on Fig. 1.
Isotherms of the pressure P, the specific internal energy
E and the entropy S depending on the density ρ for the
tungsten plasma at different temperatures
CONCLUSIONS
Evaluations we made showed that the magnetic field
up to 10
3
Т only affects the transport properties of the
plasma, but does not change the view of the inner shells
of atoms and ions.
Isotherms of the pressure and the specific internal
energy of the tungsten plasma at the temperature range
Т = 10
7
…(2∙10
8
) K and the density range
ρ = 10…10
5
kg/m
3
are obtained. The maximum values
of relative quantum and exchange corrections to the
pressure of these plasmas are ~ 1% at 10
7
K and
10
5
kg/m
3
.
This work was supported by the Ministry of
Education and Science of the Russian Federation
(Goszadanie № 13.79.2014/K).
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Article received 15.11.2014
ПРИМЕНЕНИЕ МОДЕЛИ ТОМАСА-ФЕРМИ ДЛЯ ОПРЕДЕЛЕНИЯ ТЕРМОДИНАМИЧЕСКИХ
СВОЙСТВ ЗАМАГНИЧЕННОЙ ПЛАЗМЫ
В.В. Кузенов, С.В. Рыжков, В.В. Шумаев
Определены термодинамические и транспортные свойства термоядерной плазмы, находящейся в сильном
внешнем магнитном поле (10…1000 Тл). Для этой цели использовалась модель Томаса-Ферми для веществ с
заданным температурой и плотностью. Проанализировано действие сильного магнитного поля на
транспортные свойства плазмы и вид внутренних оболочек атомов и ионов. Построены изотермы давления,
удельной внутренней энергии и энтропии плазмы вольфрама. Учтены квантовые и обменные поправки к
давлению плазмы.
ЗАСТОСУВАННЯ МОДЕЛІ ТОМАСА-ФЕРМІ ДЛЯ ВИЗНАЧЕННЯ ТЕРМОДИНАМІЧНИХ
ВЛАСТИВОСТЕЙ ЗАМАГНІЧЕНОЇ ПЛАЗМИ
В.В. Кузенoв, С.В. Рижков, В.В. Шумаєв
Визначено термодинамічні і транспортні властивості термоядерної плазми, що знаходиться в сильному
зовнішньому магнітному полі (10...1000 Тл). Для цієї мети використовувалася модель Томаса-Фермі для
речовин із заданною температурою і щільністю. Проаналізовано дію сильного магнітного поля на
транспортні властивості плазми і вид внутрішніх оболонок атомів і іонів. Побудовані ізотерми тиску,
питомої внутрішньої енергії і ентропії плазми вольфраму. Враховані квантові і обмінні поправки до тиску
плазми.
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